ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1 Development and Simulation of Mathematical Modelling of Hydraulic Turbine Gagan Singh and D.S. Chauhan ^ttarakhand Technical University, Dehradun, (U.K), India. hod_ee@dit.edu.in 2 Uttarakhand Technical University, Dehradun, (U.K),India. pdschauhan @ gmail . com Abstract- Power system performance is affected by dynamic characteristics of hydraulic governor-turbines during and following any disturbance, such as occurrence of a fault, loss of a transmission line or a rapid change of load. Accurate modelling of hydraulic System is essential to characterize and diagnose the system response. In this article the mathematical modeling of hydraulic turbine is presented. The model is capable to implement the digital systems for monitoring and control replacing the conventional control systems for power, frequency and voltage. This paper presents the possibilities of modeling and simulation of the hydro power plants and performs an analysis of different control structures and algorithms. Key words: mathematical modeling, simulation, hydraulic turbine. I. INTRODUCTION Usually, a typical hydroelectric power plant consists of water tunnel, penstock, surge tank, hydraulic turbine, speed governor, generator and electric network. Hydrodynamics and mechanoelectric dynamics are all involved in such a nonlinear dynamic system. Generally, a hydroelectric generating unit has many different operating conditions and change in any operating condition results in small or large hydraulic transients. There are many instances of damage to penstock or hydraulic turbine which are most probably occurring due to large transients. Hydrodynamics is influenced by the performance of hydraulic turbine which depends on the characteristics of the water column feeding the turbine. These characteristics include water inertia, water compressibility and pipe wall elasticity in the penstock. Different construction of hydropower systems and different operating principles of hydraulic turbines make difficult to develop mathematical models for dynamic regime, in order to design the automatic control systems. Also, there are major differences in the structure of these models. Moreover, there are major differences due to the storage capacity of the reservoir and the water supply system from the reservoir to the turbine (with or without surge chamber). The dynamic model of the plants with penstock and surge chamber is more complicated than the run-of-the-river plants, since the water feed system is a distributed parameters system. This paper will present several possibilities for the modeling of the hydraulic systems and the design of the control system. II. MODELING OF THE HYDRAULIC SYSTEM For run-of-the river types of hydropower plants have a ©2011 ACEEE DOL01.UCSI.02.02.1 low water storage capacity in the reservoir; therefore the plant operation requires a permanent balance between the water flow through turbines and the river i 4*uflbl -J ftffauifi I *BHw Hfr - 411 !■] Figure 1: Functional block diagram of hydraulic governor-turbine system interconnected with a power system network. flow in order to maximize the water level in the reservoir for a maximum efficiency of water use. Next, we will determine the mathematical model for each component of the hydropower system. A. Mathematical modeling of Hydraulic turbine The representation of the hydraulic turbine and water column in stability studies is usually based on the following assumptions:- > The hydraulic resistance is negligible. > The penstock pipe is inelastic and the water is incompressible. > The velocity of the water varies directly with the gate opening and with the square root of the net head. > The turbine output power is proportional to the product of head and volume flow. The hydraulic turbine can be considered as an element without memory since the time constants of the turbine are less smaller than the time constants of the reservoir, penstock, and surge chamber, if exists, which are series connected elements in the system. As parameters describing the mass transfer and energy transfer in the turbine we will consider the water flow through the turbine Q and the moment M generated by the turbine and that is transmitted to the electrical generator. These variables can be expressed as non- linear functions of the turbine rotational speed N, the turbine gate position Z, and the net head H of the hydro system. Q = Q(H,N,Z) (1) M = M(H,N,Z) (2) Through linearization of the equations (1) and (2) around the steady state values, we obtain: 55 -ArACEEE ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1 &Q = 3 JL &H + *R &N + 3 JL AZ ^ 3H 3N 3z q= ajj A+ fljj w+ fljj 2 3H 3jY 3Z fl2= flj j A+ iJjJ JT+ iJjJ 2 (3) (4) Where the following notations were used: _ — _ - V _ * M r. _ iJV ** ^ — — , ?1 — , TYl — , fl — , Z — — Qti iV[] Mg Hj^ Z[] Which represent the non-dimensional variations of the parameters around the steady state values. B. The hydraulic feed system The hydraulic feed system has a complex geometrical configuration, consisting of pipes or canals with different shapes and cross-sections. Therefore, the feed system will be considered as a pipe with a constant cross-section and the length equal with real length of the studied system. In order to consider this, it is necessary that the real system and the equivalent system to contain the same water mass. Let consider m , m 2 .. .m n the water masses in the pipe zones having the lengths L,L,...,l and cross- sections A „ A^,...,A of the real V 2' ' n V 2' ' n feed system. The equivalent system will have the length L=l 1 -\-l 2 -\-...+l n and cross-section A, conveniently chosen. In this case, the mass conservation law in both systems will lead to the equation: Since the water can be considered incompressible, the flow Q. through each pipe segment with cross-section A. is identical and equal with the flow Q through the equivalent pipe Q=v.A=Qi=Yi.A^ fori=l : 2 : ... : n (6) Where v is the water speed in the equivalent pipe, and v. is the speed in each segment of the real pipe. From the mass conservation law it results: v — 1 = fr^L _ £k A T.i-i'Ai Y.i-i'Ai n The dynamic pressure loss can be computed considering the inertia force of the water exerted on the cross-section of the pipe: F; = — m. a = — L. A. p. a = —A — — (S) g dt Where L is the length of the penstock or the feed canal, A is the cross-section of the penstock, g? is the specific gravity of water (1000Kgf/m 3 ), a is the water acceleration in the equivalent pipe, and g=9.81 m/s 2 is the gravitational acceleration. The dynamic pressure loss can be expressed as: _ F[ _ £-L dv _ y-L Y.[j dQ V — Li — LZl _ — S I liJ.i dt P) Using non-dimensional variations, from (9) it results: ©2011 ACEEE DOL01.UCSI.02.02.1 4tf d _ _pn_ pLElj d {\jJ tf dn ~ ** dn ' E kr*i ' dt Or in non-dimensional form: (10) ** = -T w * ■S ^ Where T w is the integration constant of the hydropower system and the variables have the following meaning: * dD QU * dD I IlAL L J X J It must be noted that this is a simplified method to compute the hydraulic pressure loss, which can be used for run-of-the river hydropower plants, with small water head. If an exact value of the dynamic pressure is required, then the formulas presented in [8], sub-chapter 8.4 "The calculation of hydro energy potential" shall be used. Using the Laplace transform in relation (1 1), it results: h d ($) = -sT w .qlj},tmd <yfc) = - — h d (s') (13) Replacing (13) in (3) and (4) and doing some simple calculations, we obtain: hi'W) = - 7 vU) - *fc> (14) (15) The mechanical power generated by the turbine can be calculated with the relation P-^.g.Q.H, which can be used to obtain the linearized relations for variations of these values around the steady state values: Where fj? is the turbine efficiency, and g, Q, and H were defined previously. On the other hand, the mechanical power can be determined using the relation P=Mu=2?z:M.N, which can be used to obtain the linearized relations for variations of these values around the steady state values: 7i = p — m (IS) 56 Where P D m w is the steady state power generated by the turbine for a given steady state flow Q and a steady state head H Q , and N is the steady state rotational speed. Using these relations, the block diagram of the hydraulic turbine, for small variation operation around the steady state point, can be determined and is presented in Figure 2, where the transfer functions for different modules are given by the following relation: *- W = i^ T A « ^/"/ ^ = TTTT^< 14-0. . T. j. J h - 14-CL. . T^.j' (-^^-) (19) *k ACEEE ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1 For an ideal turbine, without losses, the coefficients a.. y resulted from the partial derivatives in equations (12 - 16) have the following values: a n =0.5; a 12 =a 13 =l; a 21 =1.5; a 23 =l. In this case, the transfer functions in the block diagram are given by the following relation: Table I. Variations of the time constant of the hydro system JU*> = : -jU*> = : -.ff^Cs) = — H,J S y = SO—, fl^C) = (l,5 *£-) "-w=( i -^y=s D.5."T W S (20) (21) (»> III. SIMULATION RESULTS Example. Let consider a hydroelectric power system with the following parameters: -Water flow (turbines): Q N =725 m 3 /s; -Water level in the reservoir: H N =30 m; -The equivalent cross-section of the penstock A=60m 2 ; -Nominal power of the turbine P N =178MW; -Turbine efficiency rp0.94; -Nominal rotational speed of the turbine= N=71.43rot/min; -The length of the penstock l=SL=20m; *Utf *M JUtf HfcW — J j — jiki — J 2 J *Utt| I r'r.X, P Fig. 2. The block diagram of the hydraulic turbine. It shall be determined the variation of the time constant T w w for the hydro power system. For the nominal regime, using relation (12), where Jl.=20m, the time constant of the system is: r _ "5 :; 3D 9..BL.6D = 0,82s (23) Next there is a study of the variation of the time constant due to the variation of the water flow through the turbine for a constant water level in the reservoir, H=30m, as well as the variation due to the variable water level in the reservoir for a constant flow Q=725 m 3 /s. In table I, column 3 and figure 3 a) are presented the values and the graphical variation of the time constant T w for the variation of the water flow between w 500 m 3 /s and 110 m 3 /s, for a constant water level in the reservoir, H=30m. In table 1. column 4 and figure 3 b) are presented the values and the graphical variation of the time constant T w for the variation of the water level in w ©2011 ACEEE DOL01.UCSI.02.02.1 H Q Tn(H=3Qm) Tw«^7:Smc-s 17 L 1 *x*5 I.25fi0fi5 1.445101 20 5NS.14 1,093155 1 ,i3 1 736 23 &39 r 25 Q;9:5Q5tf9 1,071075 2* 742.42 0.S403S3: .9WS9 29 6G5„e: 0.753S99S 0849473 32 G03.2I 0.0832217 0,709835 35 551,51 0,G24tf598 0,703849 J8 ~--r- r r 575344C 0.G4S2S2 the reservoir, for a constant water flow, Q=725m 3 /s. 500 6W -00 WO WO 1000 !IW 1*00 T* [»] 1,1 1 T hi W [ni| 40 Figure 3. Variation of the integral time constant TW: a) by the flow Q, b) by the water level H It can be seen from the table or from the graphs that the time constant changes more than 50% for the entire operational range of the water flow through the turbine or if the water level in the reservoir varies. These variations will create huge problems during the design of the control system for the turbine, and robust control algorithms are recommended. In figure 4 the block diagram of the turbine's power con- trol system, is presented using a secondary feedback from the rotational speed of the turbine. It can be seen from this figure that a dead-zone element was inserted in series with the rotational speed sensor in order to eliminate the feedback for ±0.5% variation of the rotational speed around the syn- chronous value. This oscillation has no significant influence on the performance of the system but would have lead to 57 ^cACEEE ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1 permanent perturbation of the command sent to the turbine gate. %E1 j- 1 ' 'LTidE Figure 4. Block Diagram of the control system for hydraulic turbines The constants of the transfer functions had been computed for a nominal regime T w =0.8s. The optimal parameters for a PI controller are: K R =10, T=0.02s. The results of the turbine simulation for different operational regimes are presented in figure 5, for a control system using feedbacks from the turbine power and rotational speed, with a dead- zone on the rotational speed channel for ±0.5% variation of the rotational speed around the synchronous value (a) Power variation with 10% around nominal value, b) Rotational speed variation for power control). ■M Figure 5 Control structure with feedbacks from turbine power and rotational speed: ©2011 ACEEE DOL01.UCSI.02.02.1 In figure 6 the variations of the turbine power (graph a) and rotational speed (graph b) for the control system a feedback from the turbine power but no feedback from the rotational speed are presented. (14 Figure 6 Control structure with only power feedback a) Power variation with 10% around nominal value b) Rotational speed variation for power control IV. CONCLUSIONS The detailed mathematical modeling of hydraulic turbine is vital to capture essential system dynamic behavior .The possibility of implementation of digital systems for monitoring and control for power, frequency and voltage in the cascade hydro power plant was discussed. The simplified mathematical models, capable to accurately describe dynamic and stationary behavior of the hydro units a developed and simulated. These aspects are compared with experimental results. Finally, a practical example was used to illustrate the design of controller and to study the system stability. Acknowledgement The authors would like to thank Professor S.P. Singh (LIT Roorkee, India) for his continuous support and valuable suggestions. 58 4? ACEEE ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011 References [1] P. Kundur. Power System Stability and Control. McGraw- Hill, 1994. [2] IEEE. Hydraulic turbine and turbine control models for system dynamic studies. IEEE Transactions on Power Systems, 7(1): 167- 179, Feb 1992. [3] Jiang, J. Design an optimal robust governor for hydraulic turbine generating units IEEE Transaction on EC 1, Vol.10, 1995, pp. 18 8- 194. [4] IEEE Working Group. Hydraulic turbine and turbine control models for system dynamic studies. IEEE Transactions on Power Syst 1992; 7:167-79. [5] Nand Kishor, R.P. Saini, S.P. Singh A review on hydropower plant models and control, Renewable and Sustainable Energy Reviews 11 (2007)776-796 [6] Asal H. P., R. Widmer, H. Weber, E. Welfonder, W. Sattinger. Simulation of the restoration process after black Out in the Swiss grid. Bulletin SEV/VSE 83, 22, 1992, pp. 27-34. [7] Weber, H., F. PriUwitz, M. Hladky, H. P. Asal. Reality oriented simulation models of power plants for restoration studies. Control Engineering practice, 9, 2001, pp. 805-811. [8] Weber, H., V Fustik, F. Prillwitz, A. Iliev. Practically oriented simulation model for the Hydro Power Plant "Vrutok" in Macedonia. 2nd Balkan Power Conference, 19.-21.06. 2002, Belgrade, Yugoslavia [9] C Henderson, Yue Yang Power Station - The Implementation of the Distributed Control System, GEC Alsthom Technical Review, Nr. 10, 1992. [10] Prillwitz E, A. Hoist, H. W. Weber (2004), "Reality Oriented Simulation Models of the Hydropower Plants in Macedonia and Serbia/Montenegro" In Proceedings of the Annual Scientific Session TU Verna, Bulgaria. ©2011 ACEEE DOL01.UCSI.02.02.1 59 ^VACEEE